Higher level analogs of Nicolas-Serre theory

Question

NICOLAS-SERRE THEORY

Let $F \in Z/2[[x]]$ be $x+x^9+x^{25}+...$, the exponents being the odd squares, and $V$ be the space spanned by the $F^k$ with $k$ odd. Nicolas and Serre define formal Hecke operators $T_n$, $n$ odd, on $Z/2[[x]]$ and show:

• 1) $V$ is stable under the $T_n$. If $n$ is not a square, then $T_{n}(F^{k})$, $k$ odd, is a sum of various $F^j$ with $j$ odd and smaller than $k$. (This comes from modular form theory for the full modular group).

• 2) The only elements of $V$ annihilated by both $T_3$ and $T_5$ are $0$ and $F$.

• 3) There are $m_{a,b}$ forming a basis of $V$ such that $m_{0,0}$ is $F$, the $m_{a,b}$ with $a$ and $b$ not both $0$ are divisible by $x^2$, and $T_3$ and $T_5$ reduce $a$ by 1 and $b$ by 1 respectively.

• 4) In the action of the algebra spanned by the $T_n$, each $T_n$ "acts as a power series in $T_3$ and $T_5$". This tells us that a certain completion of this Hecke algebra is a $2$-variable power series ring over $Z/2$ in $T_3$ and $T_5$.

• The proofs of (2)-(4) are rather technical and rely on a certain "code".

CONJECTURAL ANALOGS

I've experimentally found some analogs to the above related to modular forms of level $N$ when $N$ is 3, 5, 7, or 11. The situation when $N=3$ is particularly nice. Explicitly let $D \in Z/2[[x]]$ be $x+x^{25}+x^{49}+...$, the exponents being the squares prime to $6$, and let $V$(plus) be spanned by the $D^k$ with $k \equiv 1 (\text{mod}\ 6)$. Then:

• 1*) V(plus) is stable under the $T_n$ with $n \equiv 1(\text{mod}\ 6)$. If $n$ is a non-square and $\equiv 1 (\text{mod}\ 6)$ then $T_n(D^k)$ is a sum of $D^j$ for various $j$ with $j\equiv 1 (\text{mod}\ 6)$ and $j$ smaller than $k$. (This is a theorem, whose proof uses modular forms of level 3).

• 2*) I believe that the only elements of V(plus) annihilated by $T_7$ and $T_{13}$ are $0$ and $D$.

• 3*)I believe there are $m_{a,b}$ precisely as in 3) with $F, T_3$ and $T_5$ replaced by $D, T_7$, and $T_{13}$. (I've calculated these when $a+b$ is at most 6. For example, $m_{4,2}$ is $(D^{73})+(D^{121})+(D^{145})$).

• 4*)I believe that in the action of the algebra spanned by the $T_n$ with $n\equiv 1 (\text{mod}\ 6)$ on V(plus) each such $T_n$ acts as a formal power series in $T_7$ and $T_{13}$. Also if $n$ is 1 mod 24, then $T_n$ acts as a power series in $(T_7)^2$ and $(T_{13})^2$, and in particular $(T_1)+(T_{25})=(T_5)^2$ acts by $(T_7)^2+(T_{13})^2$+(higher order terms). (This has implications for the structure of a completion of the algebra spanned by the $T_n$ with $(n,6)=1$ on the space spanned by the D^k with $(k,6)=1)$.

QUESTION: posed to those who understand the Nicolas-Serre code. Can one use some modification of the code to establish the truth of my beliefs stated above? Any thoughts about proving these conjectures would be welcome.

Remark 1: When $N=5$ there is a rather more complicated analog of V(plus) on which the T_n with n=1 or 3 mod 8 act. Results like (1) can again be proved. Results like (2)-(4) seem to hold with $T_3$ and $T_5$ replaced by $T_3$ and $T_{11}$. And if $n$ is 1 or 9 mod 40 then $T_n$ seems to be a power series in $(T_3)^2$ and $(T_{11})^2$. In particular $(T_7)^2$ acts by $(T_3)^2+(T_{11})^2$+(higher order terms). Again there would be implications for the structure of a completed Hecke algebra.

Remark 2: When $N=3$ there are similar apparent results for V(minus), the space spanned by the $D^k$ with k=5 mod 6. And when $N=5$, there is an analog of this V(minus) with similar apparent results.

EDIT___A GENERAL FRAMEWORK

I'll place the calculations I've made into a general setting, and then explain how the case N=3 described under CONJECTURAL ANALOGS fits into this expanded framework.

Let N be an odd prime and HE the "level N shallow Hecke algebra" spanned by the T_k: Z/2[[x]]-->Z/2[[x]] with(k,2N)=1. Let F be as above and G be F(x^N). I'll adopt the notation of my question 138495--"Are these two subspaces of Z/2[[x]] the same?" In particular, M is the integral closure of Z/2[G] in Z/2(F,G) viewed as a subspace of Z/2[[x]]. M has a modular forms interpretation which shows that it (and its subspaces M(odd) and C) are stable under HE. Indeed each of these is an increasing union of finite dimensional HE stable subspaces. Each of these finite dimensional subspaces is evidently annihilated by a product of powers of maximal ideals of HE. It's known, I believe, that only finitely many of these maximal ideals appear altogether. So in particular, C is a direct sum of finitely many HE stable subspaces, C(J), with the maximal ideal J acting locally nilpotently on C(J).(One case of interest is when J=Ann(F), the annihilator of F in HE. This ideal contains T_p for all primes p other than 2 or N. When N=3,5, or 7, I'm informed that Ann(F) is the only maximal ideal that appears, so that C=C(J).)

Now let pr:Z/2[[x]]-->Z/2[[x]] be the map which removes from each h in Z/2[[x]] all terms in which N divides the exponent. Then pr(C(J)) is HE stable with J acting locally nilpotently on it. As in Nicolas-Serre one can then construct a "J-completion" of HE acting faithfully on pr(C(J)). A general question is:

WHAT IS THE STRUCTURE OF THIS J-COMPLETION?

The hope that, as in Nicolas-Serre, the J-completion is a 2 variable power series ring proves mistaken. But in some cases the following seems to hold:

There are one or more index 2 subgroups of Z/(8N)* such that when one replaces HE by the subalgebra HE# spanned by the T_k with k in one of these subgroups, and C(J) by the subspace C(J)# consisting of those h in C(J) for which all the exponents that appear are either divisible by N or lie in the subgroup, then the resulting J-completion of HE# is a 2 variable power series ring. Furthermore, in a number of these cases, the J-completion of HE appears to be isomorphic to the non-reduced ring Z/2[[x,y,z]]/(z^2).

In the examples I've looked at, N=3,5,7 and 11. For each of these N, C is the set of elements of M(odd) whose trace from Z/2(F,G) to Z/2(G) is 0, and there is a Z/2[G^2] basis Ck of C with k odd, k between 0 and 2N, given in question 138495 with nice properties. Set Dk equal to pr(Ck), and extend the definition of Dm to all odd m so that whenever m=k+2N, then Dm=(G^2)Dk. Then Dm=0 when N divides m, while the remaining Dm are a Z/2 basis of pr(C). Calculations with this basis are very convenient. I won't go into the details of the calculations now, but I'll explain how the case N=3 under CONJECTURAL ANALOGS fits into this new setting.

Suppose then that N=3 and J=Ann(F). Look at the N=3 paragraph under SOME REMARKABLE FACTS in question 138495. Since C1=F, D1=pr(F) which is the D described above. Then if r=6k+1, Dr=(G^2k)*D. But classical results show that G=D^3, so that Dr=D^r for all r that are 1 mod 6. Similarly, C5=(F^2)G, so that D5=pr(F^2)G=(D^2)G=D^5, and it follows that Dr=D^r for all r that are 5 mod 6. Thus pr(C) is just the space V spanned by the D^r with (r,6)=1.

Now as I've noted, C=C(J) in this setting. Now {1,7,13,19} is an index 2 subgroup of (Z/24)*, and a Z/2 basis of the corresponding subgroup pr(C#) of pr(C) is given by the D^k with k=1 mod 6. It follows that pr(C(J)#) is just the space V(plus) spanned by these D^k, described under CONJECTURAL ANALOGS, and we are in precisely the situation given there.

EDIT(Sept. 1)__WHAT THE COMPUTER SUGGESTS IN LEVEL 11

The 1-dimensional subspace {0,F} of C is always HE stable. When N=11 there is another HE stable 1-dimensional subspace {0,t} of C, with t as in the N=11 paragraph of my question 138495. (In fact, t=C1+C3+C5+C9+C15). One way to see HE stability is the following--t^12=FG, the reduction of the expansion of the modular form delta(z)delta(11z). So t is the reduction of the expansion of (eta(z)eta(11z))^2, and this last is the weight 2 newform for Gamma_0 (11). Now Ann(t) and Ann(F) are maximal ideals in HE. Write C(t) and C(F) for C(Ann(t)) and C(Ann(F)). I've been informed that C is the direct sum of C(t) and C(F). Here's what the computer suggests for the Ann(t)-completion of HE acting on pr(C(t)).

(*) The above completion is a 2-variable power series ring over Z/2, with an element of square 0 adjoined. More precisely the map from Z/2[[X,Y,Z]] to the Ann(t)-completion that sends X,Y and Z to T_3 +I, T_5 +I and T_7 is onto and the kernel is generated by f^2 where f=Z+X+X^2+XY+Y^2+(X^2)Y+X(Y^2)+Y^3+higher degree stuff in X and Y.

Remarks__See my question 137260--Questions(related to deformation theory?)... for what the above tells us about a very natural attempted generalization of Nicolas-Serre to level 11. I have conjectures similar to (*), supported by the computer, for the Ann(F)-completion of HE acting on pr(C(F)) when N=3,5 or 11. But N=7 is more complicated; I don't understand it at this time.

Assume now that N=11. Since T_3(t)=t, T_3+ I acts nilpotently on C(t), while T_3 acts nilpotently on C(F). So we can use the calculations made of the various T_3(Dm) as sums of various Dk's to decompose each Dm into its pr(C(t)) and pr(C(F))-components for a wide range of m, and then calculate the effect of T_3, T_5 and T_7 on each of the pr(C(t))-components.

I now make use of two index 2 subgroups of (Z/88)*, G1 and G2. The first consists of the k that are squares mod 11, and the second of the 4n+1 that are squares mod 11 and the 4n+3 that are non-squares. Use G1 and G2 to define subalgebras HE(1#) and HE(2#) of HE, as well as subspaces C(t,1#) and C(t,2#) of C(t). Using the calculations outlined in the last paragraph, I empirically find:

(A)--If we replace F by pr(t)=[1,3,5,9,15] (this is shorthand for D1+D3+D5+D9+D15), T3 and T5 by T3 +I and T_5 +I, the algebra spanned by the T_n by HE(1#), and V by pr(C(t,1#)), then 2),3) and 4) under NICOLAS-SERRE THEORY still hold.

(B)--If we replace F by [1,5,9], T3 and T5 by T5 +I and T7, the algebra spanned by the T_n by HE(2#), and V by pr(C(t,2#)), then 2),3) and 4) under NICOLAS-SERRE THEORY still hold. (I've calculated many m_(a,b) both for A and for B. For example in A, m_(1,7)=[23,31,47,71,191,223]).

Now let x,y and z be the images of T3 +I, T5 +I and T7 in the Ann(t)-completion of HE, acting on pr(C(t)). From A and B one should be able to show:

1) z^2 is a power series in x^2 and y^2. (First one shows that this is true with HE replaced by HE(1#) and C(t) by C(t,1#)).

2) More precisely f^2=0, where f=z+x+x^2+xy+y^2+(x^2)y+x(y^2)+y^3+higher degree stuff in x and y. (For this one needs to write z^2 as a power series in x^2 and y^2 to the needed accuracy. This is accomplished by seeing what (T_7)^2 does to m_(7,1), m_(5,3), m(3,5) and m(1,7)).

3) Every element of the J-completion of HE acting on pr(C(t) is a power series in x,y and z.(For the elements of the J-completion of HE(1#)(resp. HE(2#)) acting on C(t) are power series in x and y(resp. y and z). And the two subalgebras generate HE.

Putting 2) and 3) together should give the desired (conjectural) structure theorem for our Ann(t)-completion of HE.

EDIT(Sept. 10) WHAT THE COMPUTER SUGGESTS IN LEVELS 3 and 5

When N=3 or 5, J can only be Ann(F), and so C=C(J). The computer indicates that in each case there is a triple of integers {n_1,n_2,n_3} with the following properties:

I.--Let a and b be two elements of the triple and G be the index 2 subgroup of (Z/8N)* generated by a,b and the squares. Use G to define the subalgebra HE# of HE and the subspace C# of C, as under WHAT IS THE STRUCTURE OF THIS J-COMPLETION? Then 2),3) and 4) under the heading NICOLAS-SERRE THEORY hold provided we replace:

The space spanned by the T_n by HE#, V by C#, T_3 by T_a and T_5 by T_b.

In particular, the Ann(F)-completion of HE# acting on C# is a power series ring in T_a and T_b over Z/2.

II.--The Ann(F)-completion of HE acting on C is a power series ring in x=T_(n_1), y=T_(n_2) and z =T_(n_3) with a single relation f^2=0, where f=x+y+z+(higher degree stuff).

IIa.--When N=3, n_1=5, n_2=7, n_3=13 and f=x+g where g=y+z+y^3+y*z^2+z^3+(higher degree stuff in y and z.

IIb.--When N=5, n_1=3, n_2=7, n_3=11 and f=y+g where g=x+z+x^3+x*z^2+z^3+(higher degree stuff in x and z.

In support of (I), I calculated the m_(i,j) for each HE# and C# whenever i+j is 6 or less. If (I) holds, an argument similar to the one I made in the last edit, should give the remaining results.

Remark__I think there are similar results when N=11, J=Ann(F), though I haven't carried out the calculations of the m_i,j very far. But N=7 is, as I've noted, very much different.

Answer 1

When N=3, and V(plus) and D are as in my question, I now believe that I have the right "code" for monomials in V(plus) and that it should be possible to use this code to yield:

(*)-- "If HE# is the algebra spanned by the operators T_n with n=1 mod 6, and m is the ideal Ann(D) in HE#, then the m-completion of HE# acting on V(plus) is a 2-variable power series ring in T_7 and T_13."

__THE NICOLAS-SERRE CODE

__ I'll begin by describing the Nicolas-Serre code for monomials in the subspace V of Z/2[[x]] spanned by the F^k, k odd, where F=x+x^9+x^25+x^49+... If c is in N, write c as a sum of distinct powers of 2, and let c2 be the sum of the squares of these powers. If (a,b) is in NxN, the "corresponding monomial in V" is F^k where k=1+2(a2)+4(b2). We say that the code of D^k is (a,b) and refer to D^k as [a,b]. Now there is a listing in order of the elements of NxN: (0,0) /(1,0),(0,1) /(2,0),(1,1),(0,2) /(3,0),(2,1),(1,2),(0,3) /... where I've written slashes where a+b jumps. This gives a corresponding listing in order of the monomials in V: D /D^3,D^5 /D^9,D^7,D^17 /D^11,D^13,D^19,D^21 /... In Propositions 4.3 and 4.4 of their paper on the order of nilpotence, Nicolas and Serre say:

__If a>0, T_3 takes [a,b] to [a-1,b]+ a sum of earlier terms.

__T_3 takes [0,b] to a sum of [i,j], with i+j < b.

__If b>0, T_5 takes [a,b] to [a,b-1]+ a sum of earlier terms.

__T_5 takes [a,0] to a sum of [i,j] with i+j < a.

---They haven't written out their proofs in the article -- they say it's long and technical and depends on certain recurrences that they explicitly give. Once the propositions are established, a fairly simple argument shows that the completion of the Hecke algebra HE acting on V, with respect to the maximal ideal Ann(F), is a power series ring in T_3 and T_5.

--THE LEVEL 3 CODE

__Now I attach to the element (a,b) of NxN the monomial D^k in V(plus) where k is 1+6(a2)+12(b2). The "code" of D^k is (a,b) and I refer to D^k as [a,b]. As above we get a listing in order of the monomials in V(plus): D /D^7,D^13 /D^25,D^19,D^49 /D^31,D^37,D^55,D^61 /... Experimentally I find that when a+b < 7 the results of Propositions 4.3 and 4.4 quoted above hold if we replace T_3 and T_5 by T_7 and T_13. And there are recursions analogous to those used by Nicolas and Serre, coming from modular equations. Deducing the analogs to the Nicolas-Serre propositions from these recursions is no doubt "long and technical" in spades, but it should be doable, and (*) from the beginning of this answer ought then to follow.

EDIT:

I now believe I can handle my N=3 conjectures, and indeed show that a certain completed Hecke algebra identifies with Z/2[[X,Y]] with an element of square 0 adjoined. Surprisingly I only need the recursions for T_7, together with information about the action of T_13 on a certain space of higher level weight one modular forms annihilated by T_7.

Here's an outline starting with another proof of the Nicolas-Serre level one theorems. Let F in Z/2[[x]] be the reduction of delta, and V be spanned by the V^k, k odd. There is a Z/2[[X,Y]]-action on V with X and Y acting by T_3 and T_5. One wants to explicitly describe V as Z/2[[X,Y]]-module (and determine the smallest k such that (X,Y)^k annihilates any given F^n).

Nicolas and Serre use their technical Propositions 4.3 and 4.4 to do this. 4.3, based on a recursion for the T_3(F^n), shows that T_3 is onto and that the kernel is "not too big" while 4.4 does the same for T_5. Mathilde Gerbelli-Gauthier has given a clear proof of 4.3, but no good proof of 4.4 is known--the Nicolas-Serre proof is long, technical and unpublished.

I circumvent 4.4 as follows. If q is a power of 2, consider positive primitive binary forms whose discriminant is -64(q^2); their SL(2,Z)-classes form a cyclic group of order 2q under Gaussian composition. Modifying the theta series attached to such a form and reducing mod 2 yields an element of Z/2[[x]]. The modified theta series are weight 1 modular forms of level a power of 2 and it turns out that each of them is congruent mod 2 to an element of V. The reductions span a space of dimension q+1, annihilated by X. Using Gaussian composition one shows that this space is stabilized by Y and is a cyclic Z/2[[Y]]-module.

Combining this with the fact that the kernel of X is "not too large" one finds that this kernel is spanned by the reductions of modified theta series and is a direct limit of cyclic Z/2[[Y]]-modules. The rest of the argument is formal and not hard--one determines the structure of V as Z/2[[X,Y]-module, shows that each T_p acts on V by multiplication by an element of (X,Y), and concludes that the completed Hecke algebra identifies with Z/2[[X,Y]].

N=3 works out a little differently. In place of F we use D=F(x)+F(x^9). V is replaced by V(minus), spanned by the D^n with n=5 mod 6. V(minus) turns out to be a Z/2[[X,Y]]-module with X and Y acting by T_7 and T_13; the proof uses level 3 modular forms. The linear map, f(t)-->(D^2)f(D^3) identifies the space of odd power series in t with V(minus). Let A(n) correspond to D^n. Using the level 7 modular equation for F one finds:

(*)___ A(n+16)=(t^16)A(n)+(t^4)A(n+4)+(t^2)A(n+2).

This is like the recursion used to prove 4.3 (but the (t^2)A(n+2) term is absent there). The initial conditions are favorable, and Gerbelli-Gauthier's argument is easily modified to show that X is onto with "not too large" kernel. One then uses modified theta series (attached to the same binary forms as above) to show that the kernel is spanned by the reductions of such series and is a direct limit of cyclic Z/2[[Y]]-modules. We then get a complete description of V(minus) as Z/2[[X,Y]]-module and a proof that each T_p with p=1 mod 6 acts on V by multiplication by an element of (X,Y). An identical argument works for V(plus).

The T_p with p=5 mod 6 don't act on V(minus) though. So to study the full Hecke algebra one looks at the sum, W, of V(minus) and V(plus). Now (T_5)^2 does act on V(minus) and is multiplication by g^2 for a certain g in (X,Y). Now if we make W into a Z/2[[X,Y,U]]-module with X,Y and U acting by T_7, T_13 and T_5 + g(X,Y), we get a complete description of W as Z/2[[X,Y,U]]-module, and conclude that the completed Hecke algebra identifies with the quotient Z/2[[X,Y,U]]/U^2.

EDIT(9/1/15)

I write out in detail the argument sketched above for N=3 in an article: "A Hecke algebra attached to mod 2 modular forms of level 3". (arXiv.org/1508.07523)

FINAL EDIT--This is now all superseded by the results in the six arXiv articles referenced in my other (now accepted) answer. See the articles, or the brief summary of them given in the other answer.

Answer 2

Working on this topic for some time,I've come to understand the level 3 and 5 analogues to the characteristic 2 level 1 theory of modular forms developed by Nicolas and Serre. My first results were presented in three arXiv articles: Variations on a Lemma of Nicolas and Serre (1604.02622 [math.NT]), A Hecke algebra attached to mod 2 modular forms of level 3 (1508.07523 [math.NT]), A Hecke algebra attached to mod 2 modular forms of level 5 (1610.07058 [math.NT]). Along the way I gave a simplification of the level 1 theory. EDIT--More precise results relating the earlier findings to structure results for more naturally appearing Hecke algebras are found in three further arXiv postings; 1603.03910, 1612.01599 and 1703.04193. I'll start with a description of the earlier results.

In each of levels 1,3,5 one has a subspace of Z/2[[x]] having as basis certain D_k. In level 1 (resp. 3,5), k runs over the positive integers prime to 2 (resp. 6,10). The formal Hecke operators T_p, p odd and not equal to the level stabilize our space; in fact T_p (D_k) is a sum of D_j with j < k. Furthermore the completed (shallow) Hecke algebra attached to the space is a power series ring Z/2[[X,Y]] in level 1, while in levels 3 and 5 one needs to adjoin an element of square zero to this power series ring. (X and Y may be taken to be T_3 and T_5 for the first space, T_7 and T_13 for the second space, T_3 and T_7 for the third space).

Here's the definition of the various spaces. Let F be x+x^9+x^25+x^49+.... In level 1, D_k is F^k. In level 3, let G(x) be F(x^3) and D(x) be F(x)+F(x^9). Then D_1 = D , D_5 = (D^2)G and D_(k+6) = (G^2)D_k. In level 5 let G(x) be F(x^5) and D(x) be F(x)+F(x^25). Then D_1 = D, D_3 = (D^8)/G, D_7 = (D^2)G, D_9 = (D^4)G and D_(k+10) = (G^2)D_k.

The main tools in the proofs are :

1. Analogues of the Nicolas-Serre "code" and the related "Proposition 4.3". The arguments here have, at their root, ideas of Mathilde Gerbelli-Gauthier.

2. Ideal theory in Z[i] in levels 1 and 3, and in Z[root -10] in level 5. This is used to study the action of Hecke operators on certain spaces of theta series attached to binary quadratic forms, and allows one to get around the difficult "Proposition 4.4" of Nicolas and Serre.

I'll explain next the connection of the three spaces described above with modular forms of level Gamma_0 (N) where N is 1,3, and 5. Let M(odd,N) consist of those "odd" elements of Z/2[[x]] that are the mod 2 reductions of expansions at infinity lying in Z[[x]] of holomorphic modular forms of level Gamma_0 (N).(Any weight is allowed).

1.---The first space, that spanned by the D_k with (k,2)=1, is M(odd,1).

EDIT:

2.---The second and third spaces are subquotients of M(odd,3) and M(odd,5). Namely there is a Hecke-stable filtration M(odd,3)> N2 > N1 > (0) with the middle quotient N2/N1 identifying with the second of our spaces. (The two outer quotients identify with M(odd,1)). An identical result holds for M(odd,5) and the last of our spaces.

3.---Using the filtration of M(odd,3) together with my results about the Hecke algebra attached to the second space, and the Nicolas-Serre results about M(odd,1), one gets the precise structure of the shallow Hecke algebra attached to M(odd,3). Namely it is the quotient of Z/2[[t_5, t_7, t_11, t_13]] (with t_p acting by T_p) by an ideal (A^2, AC, BC) where the leading forms of A, B and C are t_5 + t_7 + t_13, t_11 and t_7. There is an entirely similar result for M(odd,5).

4.---For the results of 3. see: Generators and relations for the shallow mod 2 Hecke algebra in levels Gamma_0 (3) and Gamma_0 (5), (1703.04193 math.[NT]). These results have also been obtained by Shaunak Deo and Anna Medvedovsky using techniques from deformation theory.

5.---Let K be the subspace of M(odd,3) consisting of f fixed by U_3.Then the shallow Hecke algebra attached to K is just the reduced shallow Hecke algebra attached to our second space. So it is a power series ring in T_7 and T_13, Similarly if K is the subspace of M(odd,5) consisting of f fixed by U_5, then the shallow Hecke algebra attached to K is the reduced shallow Hecke algebra attached to our third space, and is a power series ring in T_3 and T_7.